Speaker: Lennaert van Veen
Faculty of Science, University of Ontario Institute of Technology
Oshawa, ON, Canada
Title: The tangled edge of turbulence in bursting Couette flow
Abstract: Since the publication of a landmark paper by Kawahara and Kida on the relevance of unstable periodic solutions to shear flow in 2001, the scale of dynamical systems-type computations in turbulence research has increased spectacularly. Equilibrium and periodic solutions have been computed in great spatial detail for Couette flow, pipe flow and many other geometries. One of the main goals of these computations is to explain the process of turbulent bursting in shear flows. Often, the bursting occurs in the presence of an asymptotically stable laminar flow, so that ordinary bifurcation scenarios do not offer an explanation. Instead, the current focus is on so-called ``edge states,'' i.e. saddle-type equilibria or periodic solutions that appear to live on a boundary between turbulent and laminar behaviour in phase space. In principle, we should be able to clarify the bursting process if we know the geometry of the (un)stable manifolds of such states. However, the systematic computation of these manifolds is a hard task. We will present a recently developed algorithm for the computation of unstable manifolds and its adaption to turbulent Couette flow. This algorithm uses matrix-free linear solving and comes with a strong convergence result. Initial computations indicate that the (un)stable manifolds of an edge state in turbulent Couette flow form a homoclinic tangle, an observation with far-reaching implications for our understanding of the transition to turbulence.
Joint work with Genta Kawahara and Matsumura Atsushi (Osaka University)